1. MATH Algebra Problems The key to these problems is using actual numbers. Small numbers. Most choices are eliminated quickly.

2. Which of the following statements must be true whenever n, a, b, and c are positive integers such that n a, and b > c ? A. a < n B. b – n > a – n C. b < n D. n + b = a + c E. 2n > a + b A. a < n B. b – n > a – n C. b < n D. n + b = a + c First let’s put the inequality signs pointing in the same direction: I can simplify all these to this: Now let’s substitute real numbers:

3. Which one of the following expressions has an even integer value for all integers a and c? F. 8a + 2ac G. 3a + 3c H. 2a + c J. a + 2c K. ac + a 2 F. 8a + 2ac G. 3a + 3c H. 2a + c J. a + 2c K. ac + a 2 a = 0, c = 1 (8)(0)+(2)(0)(1) = 0 (3)(0)+(3)(1) = 3 (2)(0)+1 = 1 0+(2)(1) = 2 (0)(1)+1 = 1 a = 1, c = 2 (8)(1)+(2)(1)(2) = 12 1+(2)(2) = 5 Two choices remain! Let’s change the numbers and try again …

4. 55. If x and y are real numbers such that x > 1 and y < −1, then which of the following inequalities must be true? A. x = 2, y = -2 B. C. D. E.

5. When is the statement a – b = b – a true? A. Always B. Only when a and b are opposites. C. Only when a = b D. Only when a and b are both 0. E. Never

6. For all a and b such that -1 < a < 0 and 0 < b < 1, which of the following has the LARGEST VALUE? F. b G. b 2 H. a + b J. a 2 + b 2 K. b - a

7. Considering all pairs of real numbers m and n such that and m < n, which of the following statements about m must be true? A. m > 0 B. m = 0 C. m < 0 D. m > -n E. m = -n

8. If k and n are integers greater than 1 and k! is divisible by n!, then which of the following must be true? A. B. C. D. E. k is divisible by n k and n have 1 as their only common factor k and n have 1 and 2 as their only common factors